Exercise · ZW-LINEAR-TRANSFORMATION-008 · ★★☆ Intermediate
Which of these geometric operations on the plane is NOT a linear transformation?
Choose one option.
Show answer
- C. Translating the plane by adding \((1,0)\) to every point
Explanation
The translation sends the origin to \((1,0)\), but every linear transformation fixes the origin. Rotation, reflection and projection preserve sums and multiples.
Review the concept: Linear transformation
A linear transformation \(T\) is a function between vector spaces that preserves addition and scalar multiplication.
\[T(u+v)=T(u)+T(v),\quad T(cu)=cT(u)\]
More practice problems: linear transformation
- ZW-LINEAR-TRANSFORMATION-001 Let T(x,y)=(2x,3y). What is T(1,1)?
- ZW-LINEAR-TRANSFORMATION-002 Which of these functions from ℝ^2 to ℝ^2 is linear?
- ZW-LINEAR-TRANSFORMATION-003 If T is a linear transformation, what is T(0)?
- ZW-LINEAR-TRANSFORMATION-004 Let T(x,y)=(x+y, x-y). Compute T(2,3).
- ZW-LINEAR-TRANSFORMATION-005 A linear transformation T satisfies T(1,0)=(2,1) and T(0,1)=(-1,3). Compute T(3,2).
- ZW-LINEAR-TRANSFORMATION-006 A linear transformation satisfies T(1,1)=(3,1) and T(1,-1)=(1,5). What is T(1,0)?
- ZW-LINEAR-TRANSFORMATION-007 A linear transformation satisfies T(u)=(1,2) and T(v)=(3,-1). Compute T(2u-v).
- ZW-LINEAR-TRANSFORMATION-009 Let P be the space of polynomials. Which of these functions from P to P or to ℝ is linear?
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